Question

9-Sketch the phase plane portrait (phase portrait) for the given system of differential equations. Include all your calculations (phase portrait without proper calculations wont be accepted). (5 Points) X' - x + 3y ly - 2x - 4y

Answer 1

system is, solution: The given x' 2+34 y'a = -2X – 44 2 3 where X- X Y =) x -2 -4 دی ut D -4 -2 Now we will find the eigenvalues and eigeoverfors of A A.

of a is The characteristic equation der (A-05) =0 1 - 2 3 = > -4-) 0 => (2-1) (+4) +6 2 + 3) + 2 = 0 -) (2+2) (2+1) = 0 2 = -1, -2 au ut X be the eigenveetor Corresponding (2) 012 to the eigenvalue 1=-1. АХ = 2X -) (Ana X=0 6:37:03 which implies, 274 +372 - 0

11 =) su - N/W x2 (any arbitrary Constant) choosing X2 = 4= -22m ها (م m ( ) the ut Then m 2 + becomes The corresponding eigen vector is eigenvertir () = x (say) let -610 the eigenvector Corresponding the eigenvalue = -2 (A-21)X =0 Now be to 3 =) :)6)(0) -2 -2 which implies 34 +372 =0 -224 -212 =0 -Y 24 + x2 0

uf 22 = ชา (any arbitrary Constants) - . The Corresponding eigenveetor is وه me We get the eigenveto X2 (say) as choosing 0 for nog 3 X, 2 for 2 = -2) X2 phase a potrait 2 Drawing the Since both and real so the eigenvalues origin is are negative asymptotically the stable

Che sketch of the phase potrait: (3,2) (11) ya 2= -1 D = -2